# The application of fuzzy Delphi and fuzzy inference system in supplier ranking and selection

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## Abstract

In today’s highly rival market, an effective supplier selection process is vital to the success of any manufacturing system. Selecting the appropriate supplier is always a difficult task because suppliers posses varied strengths and weaknesses that necessitate careful evaluations prior to suppliers’ ranking. This is a complex process with many subjective and objective factors to consider before the benefits of supplier selection are achieved. This paper identifies six extremely critical criteria and thirteen sub-criteria based on the literature. A new methodology employing those criteria and sub-criteria is proposed for the assessment and ranking of a given set of suppliers. To handle the subjectivity of the decision maker’s assessment, an integration of fuzzy Delphi with fuzzy inference system has been applied and a new ranking method is proposed for supplier selection problem. This supplier selection model enables decision makers to rank the suppliers based on three classifications including “extremely preferred”, “moderately preferred”, and “weakly preferred”. In addition, in each classification, suppliers are put in order from highest final score to the lowest. Finally, the methodology is verified and validated through an example of a numerical test bed.

## Keywords

Supplier selection Fuzzy Delphi method Fuzzy inference system Multiple-inputs multiple-output## Introduction

One of the important decisions which play a major role in alleviation of the total cost in manufacturing systems is the effective supplier selection (Liu and Hai 2005). The selection process of vendors would be a simple task if only one criterion is used in decision making process. However, in real situation, purchasers have to consider a range of criteria to finalize their decisions. When several criteria are used, it is necessary to specify how much each criterion affects the decision-making procedure; they can be equally weighted or weights could be varied based on the type of criteria (Yahya and Kingsman 1999). According to Tahriri and Taha (2010) studies, some difficulties regarding supplier selection are included: (1) huge variety of finished products, and thus a great need for purchasing a raw material, (2) the large number of projects in process by factories, (3) the huge fluctuations in price for purchasing the raw materials, (4) the large number of suppliers by varieties of qualitative and quantitative criteria. In the same vein, Chen-Tung and Ching-Torng (2006) studied the group multiple criteria decision-making (MCDM) approach for supplier selection problem using quantity criteria. In MCDM, a problem is affected by a number of conflicting factors in supplier selection, for which a purchasing manager must analyze the tradeoff among the several criteria. MCDM techniques support the decision-makers in evaluating a set of alternatives. Depending on purchasing situations, criteria have variable importance that necessitates the weighting process (Dulmin and Mininno 2003).

In the supplier selection context, many multi-criteria decision making methods have been applied such as fuzzy TOPSIS, analytic network process (ANP), data envelopment analysis (DEA), analytic hierarchy process (AHP), and mathematical programming (Wu and Olson 2008; Kheljani et al. 2009; Lin et al. 2011; Bhattacharya et al. 2010; Moghadam et al. 2008; Wang et al. 2009; Liu and Hai 2005; Yusuff et al. 2001; Tam and Tummala 2001; Yu and Jing 2004). Another approach that can be evaluated and applied to the supplier-selection decision making process is a technique that integrates integer programming, goal programming, stochastic method, fuzzy set theory, and fuzzy multiple attribute decision making (FMADM) into a one inclusive approach. FMADM addresses the problem of choosing an alternative from a set of alternatives that are characterized according to their attributes (Tahriri et al. 2008a). The MADM method helps the manufacturer to incorporate the uncertainty of the future resulting from multi-objectivity and introduce subjective criteria in the modeling phase. On the other hand, it requires more data, and it is usually more complex than the economic analysis. Referring to the Amindoust et al. (2012) study, which proposes a fuzzy inference system (FIS) for supplier selection, the subjectivity of decision makers’ assessments are handled and the feasibility of the proposed method is shown by utilization of an illustrative example. Junior et al. (2013) proposed a new model based on fuzzy logic to handle the various attributes associated with supplier evaluation problems. Four multi-input multiple-output (MIMO) Mamdani FIS have been proposed for supplier evaluation. The proposed model has been illustrated through a case study.

Comparison of the supplier selection methods

Method | Advantages | Disadvantages |
---|---|---|

Cost ratio | Subjectivity is reduced Flexibility | Complexity and requirement for a developed cost accounting system Performance measures (cost ratios) are artificially expressed in the same units |

Principal Component analysis (PCA) | Considers simultaneously multiple inputs and outputs without priori assignment of weights | Knowledge of advanced statistical methods is required |

Artificial neural network (ANN) | Saves a lot of time and money of system development | Lack of expertise Requires a software |

Analytic hierarchical process (AHP) | Simplicity Captures both qualitative and quantitative criteria Forces managers make trade-offs Use in both criteria comparison and individual aspects within each criterion can be tackled | Inconsistency on the method Require enumerations of all issues Require intense management involvement |

Multiple attribute utility theory (MAUT) | Purchasing professionals to formulate viable sourcing strategies Capable of handling multiple conflicting attribute | Just used for international supplier selection, where the environment is more complicated and risky |

Activity-based costing (ABC) | Categorizing costs into ABC categories and then making a selection based on the criteria selected | ABC method is the method, which is most focused on cost ABC is decided base on how frequently the activity is performed in support of these cost objects When cost categories are part of the criteria |

Total cost of ownership (TCO) | Substantial cost savings Allows various purchasing policies to be compared with one another | Complex Require extensive tracking and maintenance of cost data Requires cultural change often situation specific |

Categorical | The evaluation process is clear and systematic Inexpensive Requires a minimum Performance data | Attributes are weighted equally Subjective Imprecise |

Weighted point | Attributes are weighted by importance | Subjective Difficult to effectively consider qualitative criteria |

Fuzzy TOPSIS | Simple, rationally comprehensible concept Good computational efficiency Ability to measure the relative performance for each alternative in a simple mathematical form | A disadvantage is that its use of Euclidean distance does not consider the correlation of attributes Difficult to weight attributes and keep consistency of judgment, especially with additional attributes |

Fuzzy ANP | Selection of best suitable alternative by considering various interdependent values across the supply chain and also overcome vagueness associated with the computation | The interdependence among the factors must be analyzed first to reduce the number of pairwise comparisons, which is one of most often-mentioned disadvantage |

Fuzzy Delphi | Saving time in obtaining results Reducing number of surveys required Increases questionnaire recovery rate By applying the fuzzy theory to clarify invertible fuzziness in interviews with experts to obtain more reasonable and proper responses Achieving higher economic effectiveness in time and costs required to conduct surveys Simple calculation process, handling multi-level, multi-attribute, and multi-solution decision problems Experts can fully express their opinions, ensuring the completeness and consistency of the group opinion Lower cost, and saves survey time | Takes into account the fuzziness that can’t be avoided during the survey process Does not misinterpret experts’ original opinions and provides a true reflection of their response |

Fuzzy inference system | Interpretation capability The ease of encoding a priori knowledge | Lack of learning capabilities |

An attempt has been made in this paper to identify the most important critical factors in order to select a suitable supplier through the literature. Then, a new fuzzy set multiple criteria decision-making (FMCDM) approach is proposed by integrating fuzzy Delphi (FD) and FIS to categorize the suppliers under three classifications including “extremely preferred”, “moderately preferred”, and “weakly preferred”. Moreover, the suppliers under each group are also put in order by the highest final score to the lowest.

## Determination of the supplier selection indicators

Categorization of related works for selecting the suitable supplier

Trust | Quality | Cost | Delivery | Management and organization | Financial | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Inter-firm trust | Interpersonal trust | Product | Manufacturing | Direct cost | Indirect cost | Compliance with due time | Compliance with quantity | Environment | Performance history | Facility and technical capability | Manufacturing | Product | |

Ahire (1996) | x | x | |||||||||||

Aktepe and Ersoz (2011) | x | x | x | x | x | x | x | x | x | x | x | ||

AydIn Keskin et al (2010) | x | x | x | x | x | x | x | ||||||

Amindoust et al. (2012) | x | x | x | x | x | x | x | x | x | x | x | ||

Bello (2003) | x | x | x | x | |||||||||

Benyoucef et al. (2003) | x | x | x | x | x | ||||||||

Bross and Zhao (2004) | x | x | x | x | |||||||||

Buyukozkan and Cifci (2010) | x | x | x | x | x | x | x | ||||||

Chan and Kumar (2007) | x | x | x | x | x | x | x | x | x | ||||

Chen-Tung and Ching-Torng (2006) | x | x | x | ||||||||||

Dickson (1966) | x | x | x | x | x | x | x | x | x | x | x | ||

Dyer et al. (1996) | x | x | x | x | |||||||||

Tahriri et al. (2008b) | x | x | x | x | x | x | x | x | x | x | x | x | x |

Ghodsypour and O’Brien (1998) | x | x | x | x | x | ||||||||

Handfield et al. (2002) | |||||||||||||

Junior et al. (2013) | x | x | x | x | x | x | x | x | x | x | x | ||

Liu and Hai (2005) | x | x | x | x | x | x | x | ||||||

Maani (1989) | x | x | |||||||||||

Mafakheri et al. (2011) | x | x | x | x | x | x | |||||||

Milgate (2001) | x | x | |||||||||||

Ordoobadi (2009) | x | x | x | x | |||||||||

Przewosnik et al. (2006) | x | x | x | x | x | x | x | x | x | ||||

Shen and Yu (2009) | x | x | x | x | x | ||||||||

Tam and Tummala (2001) | x | x | x | x | x | x | |||||||

Weber et al. (1991) | x | x | |||||||||||

Yahya and Kingsman (1999) | x | x | x | x | x | ||||||||

Yu and Jing (2004) | x | x | x | x | x | x | |||||||

Zhang et al. (2003) | x | x |

## Development of supplier evaluation and selection

The success supplier selection procedure

Step 1: | Defining aspects and criteria |

Step 2: | Construction of hierarchical structure |

Step 3: | Using the fuzzy Delphi method to evaluate the importance weight of each aspect based on the group decision makers |

Step 4: | Determine the importance weights for each aspect based on the linguistic scales |

Step 5: | Transfer linguistic terms of positive triangular fuzzy numbers |

Step 6: | Aggregate the mean of fuzzy weights for each aspect ( |

Step 7: | Rank the aspect based on the |

Step 8: | Design the membership function |

Step 9: | Design the fuzzy rule set |

Step 10: | Fuzzy operator |

Step 11: | Defuzzification |

Step 12: | Calculate the overall mean of defuzzification of fuzzy rule base evaluation rating ( |

### Fuzzy Delphi method

#### Defining aspects and criteria for hierarchical structure

Numerous criteria can be considered in a multi-criteria evaluation problem. These criteria should be identified under each of the *k* criteria (*C*_{1}, *C*_{2},…, *C*_{ h }, *C*_{h+1},…, *C*_{ k }) considering the specific requirements of the problem. The criteria can be classified into two categories: (1) subjective criteria, *C*_{1}, *C*_{2},…,*C*_{ h }; these criteria have a linguistic/qualitative definition; (2) objective criteria, *C*_{h+1}, *C*_{h+2},…*,C*_{ k }; these criteria are defined in monetary/quantitative terms. Based on the aspects these criteria were used to construct the hierarchy of the primary model.

#### Fuzzy Delphi method to adjust the consensus condition

Fuzzy Delphi method derived from fuzzy set theory and traditional Delphi technique is proposed by Ishikawa (1993). Noorderhaben (1995) suggested that the solution to the Fuzziness of common understanding, based on the expert’s opinions, can be performed by applying the FDM to a group decision. The application of the FDM forms a set of weights for a variety of criteria. For assessing and evaluating the performance rating, comprising the importance and appropriateness of linguistic variation, the concept of triangular fuzzy number and linguistic variables are used. Although Delphi is an expert opinion survey method having three features: Anonymous response, Iteration and controlled feedback and finally the statistical group response. Some weaknesses manifested, that block forecasting values to converge. Thus, repetitive surveys are required to perform the action which leads to needing much more time and cost (Ishikawa 1993; Wang 2008).

Delphi method provides easy understanding of the group opinions through the twice provision of the questionnaire. Since FDM integrates the fuzzy theory with the FDM, it provides the user with the advantage of Delphi method and reduction of the questionnaire time and cost (Hsu 2010; Yu-Feng 2008).

The triangular membership functions and the fuzzy theory are applied in this paper to solve the group decisions and screen the attributive factors of the first stage by using FDM. Applying the fuzzy theory can solve and evaluate the fuzziness of common understanding of experts on a variety of scales.

The FDM steps are as follows:

*Collecting opinions of decision group*the expert’s opinions are described by linguistic terms, which can be expressed in triangular fuzzy numbers, to make the consensus of the experts consistent. A committee of

*n*decision-makers (

*D*

_{1},

*D*

_{2},…,

*D*

_{ n }) evaluates the

*k*criteria and assigns a suitable weight to each one based on their importance. The committee uses linguistic weighting variables in their assessment (Zadeh 1975). Given weights are very low (

*VL*), low (

*L*), medium (

*M*), high (

*H*), and very high (

*VH*) importance. The linguistic weighting variables and the linguistic scale variables can be transformed into triangular fuzzy numbers. The membership functions for important weights are shown in Fig. 2 and Table 4, respectively.

linguistic variables and fuzzy numbers for the importance weight

Linguistic variables | Fuzzy numbers |
---|---|

Very low (VL) | (0.0, 0.0, 0.3) |

Low (L) | (0.0, 0.3, 0.5) |

Medium (M) | (0.2, 0.5, 0.8) |

High (H) | (0.5, 0.7, 1.0) |

Very high (VH) | (0.7, 1.0, 1.0) |

*Setting up triangular fuzzy numbers* the triangular fuzzy number of each factor is calculated, evaluated, and obtained based on the value given by experts.

*Aggregation of experts’ opinion*in this step, FDM is used to aggregate the decision makers’ given weights to each criteria and obtain a mean fuzzy weight under consensus condition. To this end, fuzzy numbers operations are utilized to achieve the mean of weight (

*W*

_{ t }) and transfer linguistic terms to positive trapezoidal fuzzy numbers. Suppose that

*W*

_{ t }is the linguistic weight given to subjective criteria

*C*

_{1},

*C*

_{2},…,

*C*

_{k−s}, and objective criteria

*C*

_{k−s+1},…,

*C*

_{ k }by decision maker

*D*

_{ j }. The aggregation is formulated as below:

*D*

_{ ij }is the overall average weighting valuation of alternative

*i*under criterion

*j*over

*m*assessors.

*D*

_{ ij }as a fuzzy number can be represented by the triangular membership.

*j*element is ${\stackrel{~}{W}}_{t}\phantom{\rule{0.277778em}{0ex}}=({a}_{t},{b}_{t},{c}_{t}),$

*t*= 1, 2,…,

*k*. Among which

### Fuzzy inference system

#### Fuzzifier

*X*) and the output an interval between 0 and 1 is a fuzzy degree in the qualifying linguistic set [${\mathit{\mu}}_{\stackrel{~}{A}}$(

*x*)]. By fuzzification the crisp input values are transformed into fuzzy sets. A fuzzy set $\stackrel{~}{A}$ = (

*a*,

*b*,

*c*,

*d*) in a domain

*X*is called a normal fuzzy set if $\Leftarrow $ x

_{i}є

*X*, ${\mathit{\mu}}_{\stackrel{~}{A}}$(

*x*

_{ i }) = 1. A fuzzy number is a fuzzy set in the domain of discourse ℝ (ℝ is a set of real numbers) that is convex and normal. A fuzzy set $\stackrel{~}{A}$ in the domain ℝ is called a fuzzy trapezoidal number, with a core [

*b*,

*c*], the left width

*δ*= b−a, the right width

*β*= d−c, if its membership function exhibits the shape represented by Eq. 5 and Fig. 3, (Keufmann 1991):

*a*≤

*b*≤

*c*≤

*d*and

*a*,

*b*,

*c*,

*d*є ℝ. A fuzzy trapezoidal number $\stackrel{~}{A}$ can be characterized by the ordered quadruple $\stackrel{~}{A}=(a,b,c,d)$, and can be interpreted as a fuzzy quantity “

*x*is approximately in the interval [

*b*,

*c*]”, i.e., as approximate, fuzzy interval. Constants “

*c*” and “

*d*” are the lower and upper bounds of the available area for the evaluation data. These constants reflect the fuzziness of the evaluation data. Fuzzy set is a class with a continuum of grades of membership. Let

*X*be a nonempty set, a domain of discourse where

*X*= {

*x*

_{1},

*x*

_{2},…,

*x*

_{n}}. A fuzzy (sub) set $\stackrel{~}{A}$ of a domain of discourse X is a set of ordered pairs

*X*→ [0, 1], in an accepting mathematical notation. A membership function ${\mathit{\mu}}_{\stackrel{~}{A}}$(

*x*

_{ i }) gives the membership degree of a generic element

*x*

_{ i }to the fuzzy set $\stackrel{~}{A}$. Trapezoidal fuzzy numbers are the most widely used forms of fuzzy numbers because they can be handled arithmetically and interpreted intuitively. Therefore, trapezoidal fuzzy numbers are used in this study. The MIMO FIS is used. The values of the inputs and multi separated output variables are measured and transferred to the range of the corresponding universe of discourse, which converts them into associated values.

The linguistic weighting terms of input variables

Description | Assigned weights |
---|---|

Low importance (LI) | (0, 0, 20, 30) |

Moderate importance (MI) | (20, 30, 45, 55) |

High importance (HI) | (45, 55, 70, 80) |

Extreme importance(EI) | (70, 80, 100, 100) |

The linguistic weighting terms of output variables

Description | Assigned weights |
---|---|

Weakly preferred (WP) | (0, 0, 30, 35) |

Moderately preferred (MP) | (30, 35, 65, 70) |

Extremely preferred (EP) | (65, 70, 100, 100) |

#### Fuzzy rule set

After determining the fuzzification, the input data with the conditions of the rules are determined and evaluated. There is a degree of membership for each linguistic term that applies to that input variable. The essential part of fuzzy rule based systems (FRBSs) is a set of IF–THEN linguistic rules, that have the general form “IF A THEN B” where *A* and *B* are (collections of) propositions containing linguistic variables.

*i*rules (

*R*

_{ i }) each with

*k*premises in a system, the

*i*th rule has the following form.

*X*represents the crisp inputs to the rule and

*A*

_{ i }and

*B*

_{ i }are linguistic variables, also

*X*

_{i1}to

*X*

_{ ik }and

*Y*being the input and output variables for regression respectively. The minimum value of the input variables’ membership values is based on the membership value of the control action of each rule. The number of FLs used in controlling the system using fuzzy control is represented by

*N*is the total number of rules required in controlling the system,

*m*the number of the sets of rules using one set of variables,

*n*the number of input variables used in a set of rules and

*L*the number of fuzzy sets (labels) in an input (

*i*) variable.

*α*and

*β*. The rules for the second, third input variables are using rules

*α*,

*β*as shown in Tables 7, 8. If the number of aspect input variables is great, the fix rules based on the input number of variables can be applied. This method can help the expert to design the rules easily, and precisely provide the required response and more because the number of rules of each aspect is limited.

Fuzzy if–then *α* rule

Input 2 | Input 1 | |||
---|---|---|---|---|

LI | MI | HI | EI | |

LI | LI | LI | MI | MI |

MI | LI | MI | MI | HI |

HI | MI | MI | HI | HI |

EI | MI | HI | HI | EI |

Fuzzy if–then *β* rule

Rule no. | Fuzzy input variables | Fuzzy output variables | Rule no. | Fuzzy input variables | Fuzzy output variables | Rule no. | Fuzzy input variables | Fuzzy output variables | Rule no. | Fuzzy input variables | Fuzzy output variables | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | ||||||||

1 | LI | LI | LI | LI | 17 | MI | LI | LI | LI | 33 | HI | LI | LI | MI | 49 | MI | LI | LI | LI |

2 | LI | MI | LI | LI | 18 | MI | MI | LI | MI | 34 | HI | MI | LI | MI | 50 | MI | MI | LI | MI |

3 | LI | HI | LI | LI | 19 | MI | HI | LI | MI | 35 | HI | HI | LI | MI | 51 | MI | HI | LI | MI |

4 | LI | EI | LI | LI | 20 | MI | EI | LI | MI | 36 | HI | EI | LI | MI | 52 | MI | EI | LI | MI |

5 | LI | LI | MI | LI | 21 | MI | LI | MI | MI | 37 | HI | LI | MI | MI | 53 | MI | LI | MI | MI |

6 | LI | MI | MI | MI | 22 | MI | MI | MI | MI | 38 | HI | MI | MI | MI | 54 | MI | MI | MI | MI |

7 | LI | HI | MI | MI | 23 | MI | HI | MI | MI | 39 | HI | HI | MI | HI | 55 | MI | HI | MI | MI |

8 | LI | EI | MI | MI | 24 | MI | EI | MI | MI | 40 | HI | EI | MI | HI | 56 | MI | EI | MI | MI |

9 | LI | LI | HI | MI | 25 | MI | LI | HI | MI | 41 | HI | LI | HI | HI | 57 | MI | LI | HI | MI |

10 | LI | MI | HI | MI | 26 | MI | MI | HI | MI | 42 | HI | MI | HI | HI | 58 | MI | MI | HI | MI |

11 | LI | HI | HI | MI | 27 | MI | HI | HI | HI | 43 | HI | HI | HI | HI | 59 | MI | HI | HI | HI |

12 | LI | EI | HI | MI | 28 | MI | EI | HI | HI | 44 | HI | EI | HI | HI | 60 | MI | EI | HI | HI |

13 | LI | LI | EI | MI | 29 | MI | LI | EI | HI | 45 | HI | LI | EI | HI | 61 | MI | LI | EI | HI |

14 | LI | MI | EI | HI | 30 | MI | MI | EI | HI | 46 | HI | MI | EI | HI | 62 | MI | MI | EI | HI |

15 | LI | HI | EI | HI | 31 | MI | HI | EI | HI | 47 | HI | HI | EI | HI | 63 | MI | HI | EI | HI |

16 | LI | EI | EI | EI | 32 | MI | EI | EI | HI | 48 | HI | EI | EI | EI | 64 | MI | EI | EI | EI |

#### Fuzzy operator

#### Defuzzification

*X*

^{*}is the defuzzified output,

*µ*

_{ i }(

*x*) the aggregated membership function and

*x*the output variable. The only disadvantage of this method is that it is computationally difficult for complex membership functions.

### Fuzzy Delphi multi attribute decision making

*W*

_{ i }represents aspect

*i*, and

*µ*

_{ i }(

*x*) the aggregated membership function, and

*x*the output variable. Final supplier selection results are categorized in three classifications including weakly preferred if the score is in the range of 0–35 %, moderately preferred if the score is in the range of 30–70 %, and extremely preferred if the score is in the range of 65–100 %.

### Description of the proposed model

## Illustrative example

List of the properties of attributes

No. | Fuzzy output variables | Fuzzy input variables | |
---|---|---|---|

1 | Trust ( | 1.1 | Inter-firm trust ( |

1.2 | Interpersonal trust ( | ||

2 | Quality ( | 2.1 | Product ( |

2.2 | Manufacturing ( | ||

3 | Cost ( | 3.1 | Direct cost ( |

3.2 | Indirect cost ( | ||

4 | Delivery ( | 4.1 | Compliance with due time ( |

4.2 | Compliance with quality ( | ||

5 | Management and organization ( | 5.1 | Environment ( |

5.2 | Performance history ( | ||

5.3 | Facility and technical capability ( | ||

6 | Financial ( | 6.1 | Manufacturing ( |

6.2 | Product ( |

*D*1,

*D*2,

*D*3,…,

*D*10 utilize the weight to assess the importance of each aspect. Six aspects are considered: trust (

*A*

_{1}), quality (

*A*

_{2}), cost (

*A*

_{3}), delivery (

*A*

_{4}), management and organization (

*A*

_{5}), and financial (

*A*

_{6}). It uses a FDM to adjust the fuzzy weighting by every expert to achieve the consensus condition, which obtains the important weight of the criteria by using the ten decision-makers; it is shown as Table 10.

The importance of the decision aspect

Aspects | | | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|---|

| H | VH | M | VH | L | H | M | H | L | H |

| M | H | M | M | VL | H | H | M | L | L |

| VH | M | M | VH | M | M | H | L | M | M |

| M | H | L | H | VL | L | M | H | L | L |

| M | H | L | H | H | VL | L | L | L | M |

| H | M | L | M | VH | VL | L | M | M | L |

The average linguistic rating of aspects for supplier selection and evaluation model

Aspects | | | … | | | Fuzzy mean = ΣD | Values |
---|---|---|---|---|---|---|---|

| (0.5, 0.7, 1.0) | (0.7, 1.0, 1.0) | … | (0.0, 0.3, 0.5) | (0.5, 0.7, 1.0) |
| 0.626 |

| (0.2, 0.5, 0.8) | (0.5, 0.7, 1.0) | … | (0.0, 0.3, 0.5) | (0.0, 0.3, 0.5) |
| 0.480 |

| (0.7, 1.0, 1.0) | (0.2, 0.5, 0.8) | … | (0.2, 0.5, 0.8) | (0.2, 0.5, 0.8) |
| 0.580 |

| (0.5, 0.7, 1.0) | (0.2, 0.5, 0.8) | … | (0.2, 0.5, 0.8) | (0.0, 0.3, 0.5) |
| 0.453 |

| (0.2, 0.5, 0.8) | (0.5, 0.7, 1.0) | … | (0.0, 0.3, 0.5) | (0.0, 0.3, 0.5) |
| 0.410 |

| (0.2, 0.5, 0.8) | (0.5, 0.7, 1.0) | … | (0.0, 0.3, 0.5) | (0.2, 0.5, 0.8) |
| 0.430 |

Ranking fuzzy aspect based on the *α*-cut set

Rank | Aspects | Values |
---|---|---|

1 | | H ( |

2 | | H ( |

3 | | H ( |

4 | | H ( |

5 | | H ( |

6 | | H ( |

The six aspects ranking are representative and have been studied by many scholars. The six fuzzy numbers based on six aspects are ranked as it is shown in Table 11: *A*_{5} ≺ *A*_{6} ≺ *A*_{4} ≺ *A*_{2} ≺ *A*_{3} ≺ *A*_{1}.

Summary of the basic fuzzy rule based system

No. aspect | Aspect name | No. of inputs | Rule name | No. of rules | No. of outputs | Conjunction operator | Aggregation operator | Inference model |
---|---|---|---|---|---|---|---|---|

1 | Trust | 2 | | 16 | 1 | Min | Max | Mandani |

2 | Quality | 2 | | 16 | 1 | Min | Max | Mandani |

3 | Cost | 2 | | 16 | 1 | Min | Max | Mandani |

4 | Delivery | 2 | | 16 | 1 | Min | Max | Mandani |

5 | Management and organization | 3 | | 64 | 1 | Min | Max | Mandani |

6 | Financial | 2 | | 16 | 1 | Min | Max | Mandani |

The rule numbers of two, three input criteria of six aspects are as follows: *α* rule will be used for two-input criteria (*N* _{2} ^{ α } = 4^{2)}; and *β* rule will be used for three-input criteria (*N* _{2} ^{ β } = 4^{3}).

*A*

_{1}) output criteria have two input variables included “Interfirm trust (C

_{11})”, and “Interpersonal trust (C

_{12})”. These input variables have been used as two trapezoid membership function as shown in Table 12 and Fig. 7. The next step is the formulation of the FLs. This collection of FLs approximately represents the human thinking in the decision-making process. These rules in the case of multi-input-single-output systems (MISO) are designed based on the input number variable. Rules are presented in Tables 6 and 7. Table 7 illustrates a sample of the generated FLs (

*α*= 16) for trust aspect with two input variables.

Figure 7, shows the fuzzy sets inputs (*C*_{11}), and (*C*_{12}) after implication. In the next step, all output decisions are based on testing all rules in a FIS. The rules (*R*_{ i }) must be combined in some manner in order to make a decision. The aggregation operation is used to calculate the combined fuzzy set outputs of each rule into a single fuzzy set. The output of the aggregation process is one fuzzy set for each output variable.

*C*11) and interpersonal trust (

*C*12) are applied as input variables and trust (

*A*1) as an output variable or aspect. Having verified the rules, 16 in number, which are obtained from the input number of each aspect “2”, it will be evident that the output value (trust) increases similar to the results of the input values [interfirm trust (

*C*11) and interpersonal trust (

*C*12)]. Three rules which have been placed together to show the combination and aggregation of each rule into a single fuzzy set value “output (

*A*

_{1}) = 69.3” are illustrated in Fig. 8. Finally, these fuzzy outputs need to be converted into a scalar output quantity so that the nature of the action to be performed can be determined by the center of gravity method as illustrated in Eq. 12.

*A*

_{1}) is shown in Fig. 9. Two input variables [interfirm trust (

*C*

_{11}) and interpersonal trust (

*C*

_{12})] and one output variable [trust (

*A*

_{1})] varies between 0 and 100. It can be seen that as the input values of interfirm trust (

*C*

_{11}) and interpersonal trust (

*C*

_{12}) are increased, the output value of the trust of success in supplier selection is also increased.

Final score of supplier “F” evaluation

Fuzzy input | Fuzzy output | Weight of aspect | Overall | |||
---|---|---|---|---|---|---|

Name | Value | Name | Value | Name | Value | |

| 73 | | 69.3 | WA | 0.626 | 43.38 |

| 77 | |||||

| 79 | | 76.4 | WA | 0.480 | 36.67 |

| 83 | |||||

| 86 | | 87.6 | WA | 0.580 | 50.80 |

| 82 | |||||

| 80 | | 87.6 | WA | 0.453 | 39.68 |

| 85 | |||||

| 74 | | 86.6 | WA | 0.410 | 35.50 |

| 84 | |||||

| 84 | |||||

| 85 | | 87.6 | WA | 0.430 | 37.66 |

| 89 | |||||

Final result of supplier selection = 81.81 % |

Final score and ranking of 12 suppliers

Overall | Final score (%) | Ranking | ||||||
---|---|---|---|---|---|---|---|---|

| | | | | | |||

Supplier (A) | 20.66 | 18 | 7.59 | 14.54 | 13.36 | 5.33 | 26.68 | 11 |

Supplier (B) | 49.45 | 33.26 | 49.01 | 34.60 | 30.01 | 31.82 | 76.59 | 3 |

Supplier (C) | 23.47 | 22.12 | 21.75 | 16.98 | 13.36 | 23.47 | 40.67 | 7 |

Supplier (D) | 21.72 | 18 | 16.47 | 12.86 | 17.63 | 13.80 | 33.73 | 9 |

Supplier (E) | 8.20 | 18 | 19.14 | 15.71 | 12.34 | 13.80 | 29.27 | 10 |

Supplier (F) | 43.38 | 36.67 | 50.80 | 39.68 | 35.50 | 37.66 | 81.81 | 1 |

Supplier (G) | 8.20 | 7.44 | 7.83 | 5.61 | 15.37 | 14.19 | 19.68 | 12 |

Supplier (H) | 47.82 | 30 | 42.92 | 37.01 | 27.71 | 26.87 | 71.28 | 4 |

Supplier (I) | 51.14 | 30 | 49.1 | 35.78 | 35.13 | 37.66 | 80.14 | 2 |

Supplier (J) | 20.65 | 30 | 64.62 | 33.52 | 30.13 | 23.73 | 57.94 | 6 |

Supplier (K) | 39.12 | 33.55 | 36.25 | 28.31 | 28.41 | 29.79 | 65.60 | 5 |

Supplier (L) | 23.47 | 15.84 | 33.75 | 14.90 | 12.34 | 16.12 | 39.08 | 8 |

Ranking and evaluating the suppliers based on weakly, moderately, and extremely preferred

Weakly preferred | Moderately preferred | Extremely preferred |
---|---|---|

S | S | S |

## Conclusion

The main contribution of this paper is the identification of the important criteria for selecting and evaluating the best supplier. The six aspects and thirteen criteria for supplier selection model are proposed. The six aspects are including trust, quality, cost, delivery, management and organization, financial; in which “trust” and “cost” are ranked as the top two aspects. The second contribution is the development of a multi-criteria decision making model for evaluating the criteria and selecting the appropriate supplier. This model is successfully developed by integrating the FDM and FIS methods. The rules for FIS are designed based on the number of input variables of each aspect. The number of rules for each aspect, is calculated separately based on each rule name including *α*, *β*. *α* has 16, *β* has 64, which together, there are 80 rules. Finally, the developed model is tested based on numerical test bed example with 12 suppliers. The results confirmed the model feasibility and ability to assist decision makers for examining the strengths and weaknesses of supplier by comparing them with appropriate aspects and criteria.

## Notes

### Acknowledgment

The authors would like to acknowledge the financial support provided by the Malaysian Ministry of Higher Education (MOHE) under the High Impact Research Grant (Grant No. UM.C/HIR/MOHE/ENG/35 (D000035- 16001)).

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